Tomographic imaging is a well-known imaging method wherein an object is interrogated by a set of illuminating rays that are transmitted through the object along a set of paths. As each ray passes through the object, it is attenuated based on the structure of the object along its respective path. The relative orientation of the object and the radiation source is repeatedly changed so that a complete data set of the attenuation characteristics of the structure of the object is obtained. Once scanning is completed, image reconstruction techniques are used to estimate the structure of the sample object from the resultant data set.
Unfortunately, an inordinate number of measurements are typically required to enable inference of the underlying structure of the object with adequate resolution. As a result, there has been a great deal of effort devoted over recent years to reducing the number of measurements required through improved sampling strategies.
Compressive sampling is one such approach. Compressive sampling is a signal processing technique for efficiently acquiring and reconstructing a signal by finding solutions to a system of linear equations that has more unknowns than equations (i.e., estimating a number of signal values from a number of measurements where the number of signal values is greater than the number of measurements). In order to find a solution to such a system, extra constraints, such as sparsity (i.e., assuming the signal contains many coefficients close to or equal to zero), are imposed on the system. Adding such a constraint allows only solutions having a small number of nonzero coefficients, which enables an entire signal to be determined from relatively few measurements.
In general, such an approach takes advantage of the redundancy in many signals. As a result, the measurements need to be maximally sensitive to as many of the basis images as possible and images that differ in a small number of basis images must be as distinguishable as possible in measurement space. The optimal measurements are, therefore, as unstructured as possible in terms of their sensitivity to basis images but should also maximally separate sparse representations in measurement space.
While compressive sampling represented a significant advance in the field of tomographic imaging, the reliance on completely random measurements that are spread out where the object is sparse suggests measuring different linear combinations of all coefficients. Unfortunately, tomographic measurements are inherently confined to a line in physical-location space. Further, most tomographic systems have the property that reducing the number of measurements implies reducing the number of angular samples or line integrals acquired, as opposed to measuring linear combinations of line integrals. This constraint, therefore, closely ties tomographic system design to classical sampling theory.